# Properties

 Label 100.9.d.b Level $100$ Weight $9$ Character orbit 100.d Analytic conductor $40.738$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,9,Mod(99,100)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("100.99");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$40.7378610061$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 19x^{2} + 100$$ x^4 - 19*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 4) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1) q^{2} - 8 \beta_1 q^{3} + ( - 2 \beta_{3} + 56) q^{4} + (8 \beta_{3} - 1248) q^{6} + 112 \beta_1 q^{7} + ( - 368 \beta_{2} - 144 \beta_1) q^{8} + 3423 q^{9}+O(q^{10})$$ q + (-b2 + b1) * q^2 - 8*b1 * q^3 + (-2*b3 + 56) * q^4 + (8*b3 - 1248) * q^6 + 112*b1 * q^7 + (-368*b2 - 144*b1) * q^8 + 3423 * q^9 $$q + ( - \beta_{2} + \beta_1) q^{2} - 8 \beta_1 q^{3} + ( - 2 \beta_{3} + 56) q^{4} + (8 \beta_{3} - 1248) q^{6} + 112 \beta_1 q^{7} + ( - 368 \beta_{2} - 144 \beta_1) q^{8} + 3423 q^{9} + 148 \beta_{3} q^{11} + (2496 \beta_{2} - 448 \beta_1) q^{12} + 547 \beta_{2} q^{13} + ( - 112 \beta_{3} + 17472) q^{14} + ( - 224 \beta_{3} - 59264) q^{16} + 7309 \beta_{2} q^{17} + ( - 3423 \beta_{2} + 3423 \beta_1) q^{18} - 156 \beta_{3} q^{19} - 139776 q^{21} + (23088 \beta_{2} + 14800 \beta_1) q^{22} + 18992 \beta_1 q^{23} + (2944 \beta_{3} + 179712) q^{24} + (547 \beta_{3} + 54700) q^{26} + 25104 \beta_1 q^{27} + ( - 34944 \beta_{2} + 6272 \beta_1) q^{28} + 128222 q^{29} + 544 \beta_{3} q^{31} + (24320 \beta_{2} - 81664 \beta_1) q^{32} - 184704 \beta_{2} q^{33} + (7309 \beta_{3} + 730900) q^{34} + ( - 6846 \beta_{3} + 191688) q^{36} - 347203 \beta_{2} q^{37} + ( - 24336 \beta_{2} - 15600 \beta_1) q^{38} - 4376 \beta_{3} q^{39} + 2146882 q^{41} + (139776 \beta_{2} - 139776 \beta_1) q^{42} + 474632 \beta_1 q^{43} + (8288 \beta_{3} + 4617600) q^{44} + ( - 18992 \beta_{3} + 2962752) q^{46} + 610592 \beta_1 q^{47} + (279552 \beta_{2} + 474112 \beta_1) q^{48} - 3807937 q^{49} - 58472 \beta_{3} q^{51} + (30632 \beta_{2} + 109400 \beta_1) q^{52} - 82429 \beta_{2} q^{53} + ( - 25104 \beta_{3} + 3916224) q^{54} + ( - 41216 \beta_{3} - 2515968) q^{56} + 194688 \beta_{2} q^{57} + ( - 128222 \beta_{2} + 128222 \beta_1) q^{58} - 29828 \beta_{3} q^{59} - 14746078 q^{61} + (84864 \beta_{2} + 54400 \beta_1) q^{62} + 383376 \beta_1 q^{63} + (105984 \beta_{3} - 10307584) q^{64} + ( - 184704 \beta_{3} - 18470400) q^{66} + 1221512 \beta_1 q^{67} + (409304 \beta_{2} + 1461800 \beta_1) q^{68} - 23702016 q^{69} + 9576 \beta_{3} q^{71} + ( - 1259664 \beta_{2} - 492912 \beta_1) q^{72} + 572563 \beta_{2} q^{73} + ( - 347203 \beta_{3} - 34720300) q^{74} + ( - 8736 \beta_{3} - 4867200) q^{76} + 2585856 \beta_{2} q^{77} + ( - 682656 \beta_{2} - 437600 \beta_1) q^{78} + 287536 \beta_{3} q^{79} - 53788095 q^{81} + ( - 2146882 \beta_{2} + 2146882 \beta_1) q^{82} + 4160152 \beta_1 q^{83} + (279552 \beta_{3} - 7827456) q^{84} + ( - 474632 \beta_{3} + 74042592) q^{86} - 1025776 \beta_1 q^{87} + ( - 3324672 \beta_{2} + 5446400 \beta_1) q^{88} + 83324222 q^{89} + 61264 \beta_{3} q^{91} + ( - 5925504 \beta_{2} + 1063552 \beta_1) q^{92} - 678912 \beta_{2} q^{93} + ( - 610592 \beta_{3} + 95252352) q^{94} + ( - 194560 \beta_{3} + 101916672) q^{96} + 12061901 \beta_{2} q^{97} + (3807937 \beta_{2} - 3807937 \beta_1) q^{98} + 506604 \beta_{3} q^{99}+O(q^{100})$$ q + (-b2 + b1) * q^2 - 8*b1 * q^3 + (-2*b3 + 56) * q^4 + (8*b3 - 1248) * q^6 + 112*b1 * q^7 + (-368*b2 - 144*b1) * q^8 + 3423 * q^9 + 148*b3 * q^11 + (2496*b2 - 448*b1) * q^12 + 547*b2 * q^13 + (-112*b3 + 17472) * q^14 + (-224*b3 - 59264) * q^16 + 7309*b2 * q^17 + (-3423*b2 + 3423*b1) * q^18 - 156*b3 * q^19 - 139776 * q^21 + (23088*b2 + 14800*b1) * q^22 + 18992*b1 * q^23 + (2944*b3 + 179712) * q^24 + (547*b3 + 54700) * q^26 + 25104*b1 * q^27 + (-34944*b2 + 6272*b1) * q^28 + 128222 * q^29 + 544*b3 * q^31 + (24320*b2 - 81664*b1) * q^32 - 184704*b2 * q^33 + (7309*b3 + 730900) * q^34 + (-6846*b3 + 191688) * q^36 - 347203*b2 * q^37 + (-24336*b2 - 15600*b1) * q^38 - 4376*b3 * q^39 + 2146882 * q^41 + (139776*b2 - 139776*b1) * q^42 + 474632*b1 * q^43 + (8288*b3 + 4617600) * q^44 + (-18992*b3 + 2962752) * q^46 + 610592*b1 * q^47 + (279552*b2 + 474112*b1) * q^48 - 3807937 * q^49 - 58472*b3 * q^51 + (30632*b2 + 109400*b1) * q^52 - 82429*b2 * q^53 + (-25104*b3 + 3916224) * q^54 + (-41216*b3 - 2515968) * q^56 + 194688*b2 * q^57 + (-128222*b2 + 128222*b1) * q^58 - 29828*b3 * q^59 - 14746078 * q^61 + (84864*b2 + 54400*b1) * q^62 + 383376*b1 * q^63 + (105984*b3 - 10307584) * q^64 + (-184704*b3 - 18470400) * q^66 + 1221512*b1 * q^67 + (409304*b2 + 1461800*b1) * q^68 - 23702016 * q^69 + 9576*b3 * q^71 + (-1259664*b2 - 492912*b1) * q^72 + 572563*b2 * q^73 + (-347203*b3 - 34720300) * q^74 + (-8736*b3 - 4867200) * q^76 + 2585856*b2 * q^77 + (-682656*b2 - 437600*b1) * q^78 + 287536*b3 * q^79 - 53788095 * q^81 + (-2146882*b2 + 2146882*b1) * q^82 + 4160152*b1 * q^83 + (279552*b3 - 7827456) * q^84 + (-474632*b3 + 74042592) * q^86 - 1025776*b1 * q^87 + (-3324672*b2 + 5446400*b1) * q^88 + 83324222 * q^89 + 61264*b3 * q^91 + (-5925504*b2 + 1063552*b1) * q^92 - 678912*b2 * q^93 + (-610592*b3 + 95252352) * q^94 + (-194560*b3 + 101916672) * q^96 + 12061901*b2 * q^97 + (3807937*b2 - 3807937*b1) * q^98 + 506604*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 224 q^{4} - 4992 q^{6} + 13692 q^{9}+O(q^{10})$$ 4 * q + 224 * q^4 - 4992 * q^6 + 13692 * q^9 $$4 q + 224 q^{4} - 4992 q^{6} + 13692 q^{9} + 69888 q^{14} - 237056 q^{16} - 559104 q^{21} + 718848 q^{24} + 218800 q^{26} + 512888 q^{29} + 2923600 q^{34} + 766752 q^{36} + 8587528 q^{41} + 18470400 q^{44} + 11851008 q^{46} - 15231748 q^{49} + 15664896 q^{54} - 10063872 q^{56} - 58984312 q^{61} - 41230336 q^{64} - 73881600 q^{66} - 94808064 q^{69} - 138881200 q^{74} - 19468800 q^{76} - 215152380 q^{81} - 31309824 q^{84} + 296170368 q^{86} + 333296888 q^{89} + 381009408 q^{94} + 407666688 q^{96}+O(q^{100})$$ 4 * q + 224 * q^4 - 4992 * q^6 + 13692 * q^9 + 69888 * q^14 - 237056 * q^16 - 559104 * q^21 + 718848 * q^24 + 218800 * q^26 + 512888 * q^29 + 2923600 * q^34 + 766752 * q^36 + 8587528 * q^41 + 18470400 * q^44 + 11851008 * q^46 - 15231748 * q^49 + 15664896 * q^54 - 10063872 * q^56 - 58984312 * q^61 - 41230336 * q^64 - 73881600 * q^66 - 94808064 * q^69 - 138881200 * q^74 - 19468800 * q^76 - 215152380 * q^81 - 31309824 * q^84 + 296170368 * q^86 + 333296888 * q^89 + 381009408 * q^94 + 407666688 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 19x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 29\nu ) / 5$$ (-v^3 + 29*v) / 5 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 9\nu$$ v^3 - 9*v $$\beta_{3}$$ $$=$$ $$40\nu^{2} - 380$$ 40*v^2 - 380
 $$\nu$$ $$=$$ $$( \beta_{2} + 5\beta_1 ) / 20$$ (b2 + 5*b1) / 20 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 380 ) / 40$$ (b3 + 380) / 40 $$\nu^{3}$$ $$=$$ $$( 29\beta_{2} + 45\beta_1 ) / 20$$ (29*b2 + 45*b1) / 20

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/100\mathbb{Z}\right)^\times$$.

 $$n$$ $$51$$ $$77$$ $$\chi(n)$$ $$-1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 −3.12250 + 0.500000i −3.12250 − 0.500000i 3.12250 + 0.500000i 3.12250 − 0.500000i
−12.4900 10.0000i 99.9200 56.0000 + 249.800i 0 −1248.00 999.200i −1398.88 1798.56 3680.00i 3423.00 0
99.2 −12.4900 + 10.0000i 99.9200 56.0000 249.800i 0 −1248.00 + 999.200i −1398.88 1798.56 + 3680.00i 3423.00 0
99.3 12.4900 10.0000i −99.9200 56.0000 249.800i 0 −1248.00 + 999.200i 1398.88 −1798.56 3680.00i 3423.00 0
99.4 12.4900 + 10.0000i −99.9200 56.0000 + 249.800i 0 −1248.00 999.200i 1398.88 −1798.56 + 3680.00i 3423.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.d.b 4
4.b odd 2 1 inner 100.9.d.b 4
5.b even 2 1 inner 100.9.d.b 4
5.c odd 4 1 4.9.b.b 2
5.c odd 4 1 100.9.b.c 2
15.e even 4 1 36.9.d.b 2
20.d odd 2 1 inner 100.9.d.b 4
20.e even 4 1 4.9.b.b 2
20.e even 4 1 100.9.b.c 2
40.i odd 4 1 64.9.c.b 2
40.k even 4 1 64.9.c.b 2
60.l odd 4 1 36.9.d.b 2
80.i odd 4 1 256.9.d.e 4
80.j even 4 1 256.9.d.e 4
80.s even 4 1 256.9.d.e 4
80.t odd 4 1 256.9.d.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 5.c odd 4 1
4.9.b.b 2 20.e even 4 1
36.9.d.b 2 15.e even 4 1
36.9.d.b 2 60.l odd 4 1
64.9.c.b 2 40.i odd 4 1
64.9.c.b 2 40.k even 4 1
100.9.b.c 2 5.c odd 4 1
100.9.b.c 2 20.e even 4 1
100.9.d.b 4 1.a even 1 1 trivial
100.9.d.b 4 4.b odd 2 1 inner
100.9.d.b 4 5.b even 2 1 inner
100.9.d.b 4 20.d odd 2 1 inner
256.9.d.e 4 80.i odd 4 1
256.9.d.e 4 80.j even 4 1
256.9.d.e 4 80.s even 4 1
256.9.d.e 4 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 9984$$ acting on $$S_{9}^{\mathrm{new}}(100, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 112 T^{2} + 65536$$
$3$ $$(T^{2} - 9984)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 1956864)^{2}$$
$11$ $$(T^{2} + 341702400)^{2}$$
$13$ $$(T^{2} + 29920900)^{2}$$
$17$ $$(T^{2} + 5342148100)^{2}$$
$19$ $$(T^{2} + 379641600)^{2}$$
$23$ $$(T^{2} - 56268585984)^{2}$$
$29$ $$(T - 128222)^{4}$$
$31$ $$(T^{2} + 4616601600)^{2}$$
$37$ $$(T^{2} + 12054992320900)^{2}$$
$41$ $$(T - 2146882)^{4}$$
$43$ $$(T^{2} - 35142983526144)^{2}$$
$47$ $$(T^{2} - 58160324112384)^{2}$$
$53$ $$(T^{2} + 679454004100)^{2}$$
$59$ $$(T^{2} + 13879469510400)^{2}$$
$61$ $$(T + 14746078)^{4}$$
$67$ $$(T^{2} - 232766284318464)^{2}$$
$71$ $$(T^{2} + 1430516505600)^{2}$$
$73$ $$(T^{2} + 32782838896900)^{2}$$
$79$ $$(T^{2} + 12\!\cdots\!00)^{2}$$
$83$ $$(T^{2} - 26\!\cdots\!24)^{2}$$
$89$ $$(T - 83324222)^{4}$$
$97$ $$(T^{2} + 14\!\cdots\!00)^{2}$$